Precalculus Examples

$51000 = \frac{60000}{1 + 29 e (\frac{1}{2} \ln (\frac{27}{377})) t}$

Step 1

Rewrite the equation as $\frac{60000}{1 + 29 e (\frac{1}{2}) \cdot (\ln (\frac{27}{377}) t)} = 51000$ .

$\frac{60000}{1 + 29 e (\frac{1}{2}) \cdot (\ln (\frac{27}{377}) t)} = 51000$

Step 2

Simplify $\frac{60000}{1 + 29 e (\frac{1}{2}) \cdot (\ln (\frac{27}{377}) t)}$ .

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Multiply $29 e \frac{1}{2}$ .

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Step 2.1.1.1

Combine $\frac{1}{2}$ and $29$ .

$\frac{60000}{1 + \frac{29}{2} e \cdot \ln (\frac{27}{377}) t} = 51000$

Step 2.1.1.2

Combine $\frac{29}{2}$ and $e$ .

$\frac{60000}{1 + \frac{29 e}{2} \cdot \ln (\frac{27}{377}) t} = 51000$

Step 2.1.2

Combine $\frac{29 e}{2}$ and $\ln (\frac{27}{377})$ .

$\frac{60000}{1 + \frac{29 e \ln (\frac{27}{377})}{2} t} = 51000$

Step 2.1.3

Combine $\frac{29 e \ln (\frac{27}{377})}{2}$ and $t$ .

$\frac{60000}{1 + \frac{29 e \ln (\frac{27}{377}) t}{2}} = 51000$

Step 2.1.4

Write $1$ as a fraction with a common denominator.

$\frac{60000}{\frac{2}{2} + \frac{29 e \ln (\frac{27}{377}) t}{2}} = 51000$

Step 2.1.5

Combine the numerators over the common denominator.

$\frac{60000}{\frac{2 + 29 e \ln (\frac{27}{377}) t}{2}} = 51000$

Step 2.2

Multiply the numerator by the reciprocal of the denominator.

$60000 \frac{2}{2 + 29 e \ln (\frac{27}{377}) t} = 51000$

Step 2.3

Multiply $60000 \frac{2}{2 + 29 e \ln (\frac{27}{377}) t}$ .

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Step 2.3.1

Combine $60000$ and $\frac{2}{2 + 29 e \ln (\frac{27}{377}) t}$ .

$\frac{60000 \cdot 2}{2 + 29 e \ln (\frac{27}{377}) t} = 51000$

Step 2.3.2

Multiply $60000$ by $2$ .

$\frac{120000}{2 + 29 e \ln (\frac{27}{377}) t} = 51000$

Step 2.4

Reorder factors in $\frac{120000}{2 + 29 e \ln (\frac{27}{377}) t}$ .

$\frac{120000}{2 + 29 e t \ln (\frac{27}{377})} = 51000$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 + 29 e t \ln (\frac{27}{377}), 1$

Step 3.2

Remove parentheses.

$2 + 29 e t \ln (\frac{27}{377}), 1$

Step 3.3

The LCM of one and any expression is the expression.

$2 + 29 e t \ln (\frac{27}{377})$

Step 4

Multiply each term in $\frac{120000}{2 + 29 e t \ln (\frac{27}{377})} = 51000$ by $2 + 29 e t \ln (\frac{27}{377})$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{120000}{2 + 29 e t \ln (\frac{27}{377})} = 51000$ by $2 + 29 e t \ln (\frac{27}{377})$ .

$\frac{120000}{2 + 29 e t \ln (\frac{27}{377})} (2 + 29 e t \ln (\frac{27}{377})) = 51000 (2 + 29 e t \ln (\frac{27}{377}))$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2 + 29 e t \ln (\frac{27}{377})$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{120000}{2 + 29 e t \ln (\frac{27}{377})} (2 + 29 e t \ln (\frac{27}{377})) = 51000 (2 + 29 e t \ln (\frac{27}{377}))$

Step 4.2.1.2

Rewrite the expression.

$120000 = 51000 (2 + 29 e t \ln (\frac{27}{377}))$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify by multiplying through.

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Step 4.3.1.1

Apply the distributive property.

$120000 = 51000 \cdot 2 + 51000 (29 e t \ln (\frac{27}{377}))$

Step 4.3.1.2

Multiply.

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Step 4.3.1.2.1

Multiply $51000$ by $2$ .

$120000 = 102000 + 51000 (29 e t \ln (\frac{27}{377}))$

Step 4.3.1.2.2

Multiply $29$ by $51000$ .

$120000 = 102000 + 1479000 (e t \ln (\frac{27}{377}))$

Step 4.3.2

Remove parentheses.

$120000 = 102000 + 1479000 e t \ln (\frac{27}{377})$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $102000 + 1479000 e t \ln (\frac{27}{377}) = 120000$ .

$102000 + 1479000 e t \ln (\frac{27}{377}) = 120000$

Step 5.2

Move all terms not containing $t$ to the right side of the equation.

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Step 5.2.1

Subtract $102000$ from both sides of the equation.

$1479000 e t \ln (\frac{27}{377}) = 120000 - 102000$

Step 5.2.2

Subtract $102000$ from $120000$ .

$1479000 e t \ln (\frac{27}{377}) = 18000$

Step 5.3

Divide each term in $1479000 e t \ln (\frac{27}{377}) = 18000$ by $1479000 e \ln (\frac{27}{377})$ and simplify.

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Step 5.3.1

Divide each term in $1479000 e t \ln (\frac{27}{377}) = 18000$ by $1479000 e \ln (\frac{27}{377})$ .

$\frac{1479000 e t \ln (\frac{27}{377})}{1479000 e \ln (\frac{27}{377})} = \frac{18000}{1479000 e \ln (\frac{27}{377})}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $1479000$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{1479000 e t \ln (\frac{27}{377})}{1479000 e \ln (\frac{27}{377})} = \frac{18000}{1479000 e \ln (\frac{27}{377})}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{e t \ln (\frac{27}{377})}{e \ln (\frac{27}{377})} = \frac{18000}{1479000 e \ln (\frac{27}{377})}$

Step 5.3.2.2

Cancel the common factor of $e$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{e t \ln (\frac{27}{377})}{e \ln (\frac{27}{377})} = \frac{18000}{1479000 e \ln (\frac{27}{377})}$

Step 5.3.2.2.2

Rewrite the expression.

$\frac{t \ln (\frac{27}{377})}{\ln (\frac{27}{377})} = \frac{18000}{1479000 e \ln (\frac{27}{377})}$

Step 5.3.2.3

Cancel the common factor of $\ln (\frac{27}{377})$ .

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Step 5.3.2.3.1

Cancel the common factor.

$\frac{t \ln (\frac{27}{377})}{\ln (\frac{27}{377})} = \frac{18000}{1479000 e \ln (\frac{27}{377})}$

Step 5.3.2.3.2

Divide $t$ by $1$ .

$t = \frac{18000}{1479000 e \ln (\frac{27}{377})}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $18000$ and $1479000$ .

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Step 5.3.3.1.1

Factor $3000$ out of $18000$ .

$t = \frac{3000 (6)}{1479000 e \ln (\frac{27}{377})}$

Step 5.3.3.1.2

Cancel the common factors.

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Step 5.3.3.1.2.1

Factor $3000$ out of $1479000 e \ln (\frac{27}{377})$ .

$t = \frac{3000 (6)}{3000 (493 e \ln (\frac{27}{377}))}$

Step 5.3.3.1.2.2

Cancel the common factor.

$t = \frac{3000 \cdot 6}{3000 (493 e \ln (\frac{27}{377}))}$

Step 5.3.3.1.2.3

Rewrite the expression.

$t = \frac{6}{493 e \ln (\frac{27}{377})}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = \frac{6}{493 e \ln (\frac{27}{377})}$

Decimal Form:

$t = - 0.00169823 \dots$